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Calculus II

MATH 152 Course Notes

Department. of Mathematics, SFU

Spring 2023

Copyright © 2023 Veselin Jungic and Jamie Mulholland, SFU

SELFPUBLISHED

http://www.math.sfu.caLicensed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License (the

"License"). You may not use this document except in compliance with the License. You may obtain a copy of the License athttp://creativecommons.org/licenses/by-nc-sa/4.0/. Unless required by applicable law or agreed to in writing, software distributed under the License is dis- tributed on an"AS IS"BASIS,WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License.

First printing, August 2006

Contents

Preface.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Greek Alphabet.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 IPart One: Introduction to the Integral

1Integrals.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.1 Areas and Distances

14

1.2 The Definite Integral

19

1.3 The Fundamental Theorem of Calculus

28

1.4 The Net Change Theorem

34

1.5 The Substitution Rule

39

2Applications of Integration.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.1 Areas Between Curves

48

2.2 Areas in Polar Coordinates

52

2.3 Volumes55

2.4 Volumes by Cylindrical Shells

62 IIPart Two: Integration Techniques and Applications

3Techniques of Integration and Applications.. . . . . . . . . . . . . . . . . . . 69

3.1 Integration By Parts

70

3.2 Trigonometric Integrals

76

3.3 Trigonometric Substitutions

80

3.4 Integration of Rational Functions by Partial Fractions82

3.5 Strategy for Integration

89

3.6 Approximate Integration

95

3.7 Improper Integrals

103

4Further Applications of Integration.. . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.1 Arc Length

110

4.2 Area of a Surface of Revolution

114

4.3 Calculus with Parametric Curves

118 IIIPart Three: Sequences and Series

5Infinite Sequences and Series.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.1 Sequences

126

5.2 Series133

5.3 The Integral Test and Estimates of Sums

139

5.4 The Comparison Test

143

5.5 Alternating Series

147

5.6 Absolute Convergence and the Ratio and Root Test

151

5.7 Strategy for Testing Series

156

5.8 Power Series

160

5.9 Representation of Functions as Power Series

164

5.10 Taylor and Maclaurin Series

167

5.11 Applications of Taylor Polynomials

175 IVPart Four: Differential Equations

6A First Look at Differential Equations.. . . . . . . . . . . . . . . . . . . . . . . . . . 181

6.1 Modeling with Differential Equations, Directions Fields

182

6.2 Separable Equations

187

6.3 Models for Population Growth

194 VExam Preparation

7Review Materials for Exam Preparation.. . . . . . . . . . . . . . . . . . . . . . . 199

7.1 Midterm 1 Review Package

200

7.2 Midterm 2 Review Package

207

7.3 Final Exam Practice Questions

216 VIAppendix

Bibliography.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

Articles223

Books223

Web Sites223

Index.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

PrefaceThis booklet contains the note templates for coursesMath 150/151 - Calculus Iat Simon Fraser University. Students are expected to use this booklet during each lecture by following along with the instructor, filling in the details in the blanks provided. Definitions and theorems appear in highlighted boxes.

Next to some examples you"ll see [

link to applet ]. The link will take you to an online interactive

applet to accompany the example - just like the ones used by your instructor in the lecture. The link

above will take you to the following url [Mul22] containing all the applets:

Try it now.

Next to some section headings you"ll notice a QR code. They look like the image on the right. Each one provides a link to a webpage (could be a youtube video, or access to online Sage code). For example this one takes you to the Wikipedia page which explains what a QR code is. Use a QR code scanner on your phone or tablet and it will quickly take you off to the webpage. The app "Red Laser" is a good QR code scanner which is

available for free (iphone, android, windows phone).We offer a special thank you to Keshav Mukunda for his many contributions to these notes.

No project such as this can be free from errors and incompleteness. We will be grateful to everyone who points out any typos, incorrect statements, or sends any other suggestion on how to improve this manuscript.

Veselin Jungic Jamie Mulholland

vjungic@sfu.ca j_mulholland@sfu.ca

January 13, 2023

Greek Alphabetlower

case capital name pronunciationlower case capital name pronunciationαAalpha (al-fah)νNnu (new)

βBbeta (bay-tah)ξΞxi (zie)

γΓgamma (gam-ah)o Oomicron (om-e-cron)

δ∆delta (del-ta)πΠpi (pie)

εEepsilon (ep-si-lon)ρPrho (roe)

ηHeta (ay-tah)τTtau (taw)

θΘtheta (thay-tah)υϒupsilon (up-si-lon)

ιIiota (eye-o-tah)φΦphi (fie)

κKkappa (cap-pah)χXchi (kie)

λΛlambda (lamb-dah)ψΨpsi (si)

µMmu (mew)ωΩomega (oh-may-gah)

I

1Integrals.. . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.1 Areas and Distances

1.2 The Definite Integral

1.3 The Fundamental Theorem of Calculus

1.4 The Net Change Theorem

1.5 The Substitution Rule

2Applications of Integration.. . . . . . . . . . 47

2.1 Areas Between Curves

2.2 Areas in Polar Coordinates

2.3 Volumes

2.4 Volumes by Cylindrical ShellsPart One: Introduction to the

Integral

1. IntegralsIn this chapter we lay down the foundations for this course. We introduce the two motivating

problems for integral calculus: the area problem, and the distance problem. We then define the integral and discover the connection between integration and differentiation.

14Chapter 1. Integrals1.1Ar easand Distances

(This lecture corresponds to Section 5.1 of Stewart"sCalculus.) One can never know for sure what a desertedarealooks like. (George Carlin, American stand-up Comedian, Actor and Author, 1937-2008)

BIG Question.What is the meaning of the wordarea?

Vocabulary.Cambridge dictionary:

areanoun (a) a particular part of a place, piece of land or country; (b) the size of a flat surf acecalculated by multiplying its length by its width; (c) a subject or acti vity,or a part of it. (d) (W ikipedia)- Area is a ph ysicalquantity e xpressingthe size

of a part of a surface.Example 1.1Find the area of the region in the coordinate plane bounded by the coordinate

axes and linesx=2 andy=3. Example 1.2Find the area of the region in the coordinate plane bounded by thex-axis and linesy=2xandx=3. Example 1.3Find the area of the region in the coordinate plane bounded by thex-axis and linesy=x2andx=3.

1.1 Areas and Distances 15Example 1.4Estimatethe area of the region in the coordinate plane bounded by thex-axis and

curvesy=x2andx=3.

16Chapter 1. IntegralsExample 1.5(Over- and under-estimates.)In the previous example, show that

lim n→∞Rn=9 and limn→∞Ln=9.

A more general formulation.

Ingredients: A functionfthat is continuous on a closed interval[a,b].

Letn∈N, and define∆x=b-an

Let x 0=a x

1=a+∆x

x

2=a+2∆x

x

3=a+3∆x

x n=a+n∆x=b.

Define

R

("R" stands for "right-hand", since we are using the right hand endpoints of the little rectangles.)Definition 1.1.1 - Area.TheareaAof the regionSthat lies under the graph of the continuous

functionfover and interval[a,b]is the limit of the sum of the areas of approximating rectangles

Rn. That is,

The more compactsigma notationcan be used to write this as

A=limn→∞Rn=limn→∞

n∑ i=1f(xi)! ∆x.

1.1 Areas and Distances 17

Example 1.6Find the area under the graph off(x) =100-3x2fromx=1 tox=5. From the definition of area, we haveA=limn→∞ n∑ i=1f(xi)! ∆x.Distance Problem.Find the distance traveled by an object during a certain time period if the velocity of the object is known at all times.Reminderdistance = velocity·time

18Chapter 1. IntegralsAdditional Notes:

1.2 The Definite Integral 19

1.2

The Definite Integral

(This lecture corresponds to Section 5.2 of Stewart"sCalculus.)After years of finding mathematics easy, I finally reached integral calculus and came up against

a barrier. I realized that this was as far as I could go, and to this day I have never successfully gone beyond it in any but the most superficial way. (Isaac Asimov, Russian-born American author and biochemist, best known for his works of science fiction, 1920-1992) Definition 1.2.1 - The Definite Integral.Supposefis a continuous function defined on the closed interval[a,b], we divide[a,b]intonsubintervals of equal width∆x= (b-a)/n. Let x

0=a,x1,x2, ...,xn=b

be the end points of these subintervals. Let x ∗1,x∗2,...,x∗n be anysample pointsin these subintervals, sox∗ilies in theith subinterval[xi-1,xi]. Then thedefinite integral of f from a to bis written asZ b af(x)dx, and is defined as follows: Zb af(x)dx=limn→∞n∑ i=1f(x∗i)∆x

20Chapter 1. IntegralsThe definite integral: some terminology

Z b af(x)dx=limn→∞n∑ i=1f(x∗i)∆x Z is theintegral sign

•f(x)is theintegrand

•aandbare thelimits of integration:

•a-lower limit •b-upper limit The procedure of calculating an inte gralis called integration.

•n∑

i=1f(x∗i)∆xis called aRiemann sum (named after the German mathematician Bernhard Riemann,1826-1866)

Four Facts.

(a)

If f(x)>0 on[a,b]thenZ

b af(x)dx>0.

Iff(x)<0 on[a,b]thenZ

b af(x)dx<0. (b)

F ora general function f,

Z b af(x)dx=(signed area of the region) = (area abovex-axis) - (area belowx-axis) (c) F ore veryε>0 there exists a numberN∈Nsuch that Z b af(x)dx-n∑ i=1f(x∗i)∆x for everyn>Nand every choice ofx∗1,x∗2,...,x∗n. (d)Letfbe continuous on[a,b]and leta=x01.2 The Definite Integral 21

Some facts you just have to know.

1 (a) n∑ i=1i=n(n+1)2 (b) n∑ i=1i2=n(n+1)(2n+1)6 (c) n∑ i=1i3=n(n+1)2 2 (d) n∑ i=1c=cn (e) n∑ i=1(cai) =cn∑ i=1a i (f) n∑ i=1(ai±bi) =n∑ i=1a i±n∑ i=1b

i1For visual proofs of (a) and (b) see [Gol02]: Goldoni, G. (2002)A visual proof for the sum of the first n squares

and for the sum of the first n factorials of order two. The Mathematical Intelligencer 24 (4): 67-69. You can access the

Mathematical Intelligencer through the SFU Library web site:http://cufts2.lib.sfu.ca/CJDB/BVAS/journal/

150620.

22Chapter 1. IntegralsExample 1.7EvaluateZ2

0(x2-x)dx.Example 1.8Express the limit

lim n→∞n∑ i=1(1+xi)cosxi∆x as a definite integral on the interval[π,2π].Example 1.9ProveZ2

0p4-x2dx=π.

1.2 The Definite Integral 23Theorem 1.2.1- Choosing a good sample point . ...Midpoint Rule.To approximate an

integral it is usually better to choosex∗ito be the midpointx iof the interval[xi-1,xi]: Z b af(x)dx≈n∑ i=1f(x i)∆x=∆x[f(x

1)+f(x

2)+...+f(x

n)] Recall the midpoint of an interval[xi-1,xi]is given byx i=12 (xi-1+xi).Example 1.10Use the Midpoint Rule withn=4 to approximate the integralZ 5 1dxx 2.

24Chapter 1. IntegralsTheorem 1.2.2- T woSpecial Pr opertiesof the Integral.

(a)

If a>bthenZb

af(x)dx=-Z a bf(x)dx. (b)

If a=bthenZb

af(x)dx=0.

Some More Properties of the Integral.

(a)

If cis a constant, thenZ

b acdx=c(b-a) (b) Z b a[f(x)±g(x)]dx=Z b af(x)dx±Z b ag(x)dx (c)

If cis a constant, thenZ

b acf(x)dx=cZ b af(x)dx (d) Z c af(x)dx+Z b cf(x)dx=Z bquotesdbs_dbs22.pdfusesText_28

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